T10 - Tutorial 10th. 11/17/2008 Eigenvalues and...

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Tutorial 10th. 11/17/2008 Eigenvalues and Eigenvectors
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Definition 1 { : linear transformation from z Eigenvalue : the satisfies for some x. z Eigenvector : The x associated with the eigenvalue . { Example 1: z Compute the eigenvalues and eigenvectors of the following linear transformation
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Solution to Example 1 { What we want to find is the and satisfies { Assume , then, the above equation is equivalent to { Which yields
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Solution Cont. { The existence of nonzero solutions to the above equation (1) gives out { Then, we get and . { Substitute into (1), we get where { Substitute into (1), we get where
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Definition 2 { A: an matrix. z Characteristic polynomial of A: z Characteristic equation of A: { The eigenvalues of A are the roots of the characteristic equation of A. { The eigenvector for each eigenvalue of A is the solution to { Similar matrices have the same eigenvalues.
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Definition 3 { : Linear transformation from z is diagonalizable if there exists a basis for such that L is represented with respect to S by a
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This note was uploaded on 11/13/2010 for the course SEEM SEEM2018 taught by Professor Chan during the Spring '10 term at CUHK.

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T10 - Tutorial 10th. 11/17/2008 Eigenvalues and...

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